Metric propositional neighborhood logic with an equivalence relation

Abstract

The propositional interval logic of temporal neighborhood (PNL for short) features two modalities that make it possible to access intervals adjacent to the right (modality xAy) and to the left (modality xAy) of the current interval. PNL stands at a central position in the realm of interval temporal logics, as it is expressive enough to encode meaningful temporal conditions and decidable (undecidability rules over interval temporal logics, while PNL is NEXPTIME-complete). Moreover, it is expressively complete with respect to the two-variable fragment of first-order logic extended with a linear order FO 2 răs. Various extensions of PNL have been studied in the literature, including metric, hybrid, and first-order ones. Here, we study the effects of the addition of an equivalence relation " to Metric PNL (MPNL"). We first show that the finite satisfiability problem for PNL extended with " is still NEXPTIME-complete. Then, we prove that the same problem for MPNL" can be reduced to the decidable 0-0 reachability problem for vector addition systems and vice versa (EXPSPACE-hardness immediately follows).

Bounded variability of metric temporal logic

“…Finding matching lower bound in some uncommon semantic interpretations of MTL over dense time (Section 6.4) would require using novel techniques involving channel machines with insertion errors [44,45]. Other, broader and hence potentially more interesting, open questions involve studying bounded variability for different kinds of metric temporal logics, including recent proposals for classic [17,55,56] or interval-based [8,39] real-time notations.…”

“…In fact, interval temporal logics are highly expressive, often undecidable formalisms whose detailed classification has occupied researchers up until recent years [7]. Only recently have real-time interval temporal logics been studied in some detail [9,39]. Some interval logics, such as propositional neighborhood logic, remain decidable over the reals [41].…”